Radio pulsations from the $\gamma$-ray millisecond pulsar PSR J2039-5617

The predicted nature of the candidate redback pulsar 4FGL J2039.5-56.17 was recently confirmed by the discovery of $\gamma$-ray millisecond pulsations (Clark et al. 2020, hereafter Paper I), which identify this $\gamma$-ray source as PSR J2039-5617. We observed 4FGL 2039.5-5617 with the Parkes radio telescope in 2016 and 2019. We detect radio pulsations at 1.4 GHz and 3.1 GHz, at the 2.6 ms period discovered in $\gamma$-rays, and also at 0.7 GHz in one 2015 archival observation. In all bands, the radio pulse profile is characterised by a single relatively broad peak which leads the main $\gamma$-ray peak. At 1.4 GHz we found clear evidence of eclipses of the radio signal for about half of the orbit, a characteristic phenomenon in redback systems, which we associate with the presence of intra-binary gas. From the dispersion measure DM$=24.57\pm0.03$\,pc\,cm$^{-3}$ we derive a pulsar distance of $0.9\pm 0.2$kpc or $1.7\pm0.7$kpc, depending on the assumed Galactic electron density model. Both the 1.4 GHz Parkes observations obtained in 2016 and 2019 and Swift monitoring campaign (2017-2018) show no obvious long-term changes in the pulsar radio and X-ray emission. The modelling of the radio and $\gamma$-ray light curves leads to an independent determination of the orbital inclination, and to a determination of the pulsar mass, qualitatively consistent to the results in Paper I.